Show commands:
Magma
magma: G := TransitiveGroup(26, 27);
Group invariants
| Abstract group: | $C_{13}^2:(C_3\times S_3)$ | magma: IdentifyGroup(G);
| |
| Order: | $3042=2 \cdot 3^{2} \cdot 13^{2}$ | magma: Order(G);
| |
| Cyclic: | no | magma: IsCyclic(G);
| |
| Abelian: | no | magma: IsAbelian(G);
| |
| Solvable: | yes | magma: IsSolvable(G);
| |
| Nilpotency class: | not nilpotent | magma: NilpotencyClass(G);
|
Group action invariants
| Degree $n$: | $26$ | magma: t, n := TransitiveGroupIdentification(G); n;
| |
| Transitive number $t$: | $27$ | magma: t, n := TransitiveGroupIdentification(G); t;
| |
| Parity: | $-1$ | magma: IsEven(G);
| |
| Primitive: | no | magma: IsPrimitive(G);
| |
| $\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
| |
| Generators: | $(1,16,3,21,9,23)(2,25,6,22,5,26)(4,17,12,24,10,19)(7,18,8,14,11,15)(13,20)$, $(1,6,8)(2,9,4)(3,12,13)(7,11,10)(14,16,18,20,22,24,26,15,17,19,21,23,25)$ | magma: Generators(G);
|
Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $6$: $S_3$, $C_6$ $18$: $S_3\times C_3$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 13: None
Low degree siblings
39T39, 39T40Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
| Label | Cycle Type | Size | Order | Index | Representative |
| 1A | $1^{26}$ | $1$ | $1$ | $0$ | $()$ |
| 2A | $2^{13}$ | $39$ | $2$ | $13$ | $( 1,26)( 2,14)( 3,15)( 4,16)( 5,17)( 6,18)( 7,19)( 8,20)( 9,21)(10,22)(11,23)(12,24)(13,25)$ |
| 3A1 | $3^{4},1^{14}$ | $26$ | $3$ | $8$ | $(14,18,17)(15,21,26)(16,24,22)(19,20,23)$ |
| 3A-1 | $3^{4},1^{14}$ | $26$ | $3$ | $8$ | $( 1, 4, 5)( 2,13, 8)( 3, 9,11)( 6,10, 7)$ |
| 3B1 | $3^{8},1^{2}$ | $169$ | $3$ | $16$ | $( 2,10, 4)( 3, 6, 7)( 5,11,13)( 8,12, 9)(14,22,16)(15,18,19)(17,23,25)(20,24,21)$ |
| 3B-1 | $3^{8},1^{2}$ | $169$ | $3$ | $16$ | $( 2, 4,10)( 3, 7, 6)( 5,13,11)( 8, 9,12)(14,16,22)(15,19,18)(17,25,23)(20,21,24)$ |
| 3C | $3^{8},1^{2}$ | $338$ | $3$ | $16$ | $( 1, 5, 4)( 2, 8,13)( 3,11, 9)( 6, 7,10)(14,15,24)(16,20,17)(18,25,23)(19,21,26)$ |
| 6A1 | $6^{4},2$ | $507$ | $6$ | $21$ | $( 1,26)( 2,16,10,14, 4,22)( 3,19, 6,15, 7,18)( 5,25,11,17,13,23)( 8,21,12,20, 9,24)$ |
| 6A-1 | $6^{4},2$ | $507$ | $6$ | $21$ | $( 1,26)( 2,22, 4,14,10,16)( 3,18, 7,15, 6,19)( 5,23,13,17,11,25)( 8,24, 9,20,12,21)$ |
| 13A1 | $13,1^{13}$ | $6$ | $13$ | $12$ | $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)$ |
| 13A-1 | $13,1^{13}$ | $6$ | $13$ | $12$ | $( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)$ |
| 13A2 | $13,1^{13}$ | $6$ | $13$ | $12$ | $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)$ |
| 13A-2 | $13,1^{13}$ | $6$ | $13$ | $12$ | $( 1, 6,11, 3, 8,13, 5,10, 2, 7,12, 4, 9)$ |
| 13B1 | $13^{2}$ | $9$ | $13$ | $24$ | $( 1, 7,13, 6,12, 5,11, 4,10, 3, 9, 2, 8)(14,19,24,16,21,26,18,23,15,20,25,17,22)$ |
| 13B-1 | $13^{2}$ | $9$ | $13$ | $24$ | $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,25,23,21,19,17,15,26,24,22,20,18,16)$ |
| 13B2 | $13^{2}$ | $9$ | $13$ | $24$ | $( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)(14,23,19,15,24,20,16,25,21,17,26,22,18)$ |
| 13B-2 | $13^{2}$ | $9$ | $13$ | $24$ | $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(14,24,21,18,15,25,22,19,16,26,23,20,17)$ |
| 13C1 | $13^{2}$ | $18$ | $13$ | $24$ | $( 1, 6,11, 3, 8,13, 5,10, 2, 7,12, 4, 9)(14,25,23,21,19,17,15,26,24,22,20,18,16)$ |
| 13C2 | $13^{2}$ | $18$ | $13$ | $24$ | $( 1,10, 6, 2,11, 7, 3,12, 8, 4,13, 9, 5)(14,26,25,24,23,22,21,20,19,18,17,16,15)$ |
| 13D1 | $13^{2}$ | $18$ | $13$ | $24$ | $( 1,10, 6, 2,11, 7, 3,12, 8, 4,13, 9, 5)(14,21,15,22,16,23,17,24,18,25,19,26,20)$ |
| 13D-1 | $13^{2}$ | $18$ | $13$ | $24$ | $( 1, 6,11, 3, 8,13, 5,10, 2, 7,12, 4, 9)(14,15,16,17,18,19,20,21,22,23,24,25,26)$ |
| 13D2 | $13^{2}$ | $18$ | $13$ | $24$ | $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)(14,18,22,26,17,21,25,16,20,24,15,19,23)$ |
| 13D-2 | $13^{2}$ | $18$ | $13$ | $24$ | $( 1,11, 8, 5, 2,12, 9, 6, 3,13,10, 7, 4)(14,16,18,20,22,24,26,15,17,19,21,23,25)$ |
| 26A1 | $26$ | $117$ | $26$ | $25$ | $( 1,14, 3,20, 5,26, 7,19, 9,25,11,18,13,24, 2,17, 4,23, 6,16, 8,22,10,15,12,21)$ |
| 26A-1 | $26$ | $117$ | $26$ | $25$ | $( 1,16,13,26,12,23,11,20,10,17, 9,14, 8,24, 7,21, 6,18, 5,15, 4,25, 3,22, 2,19)$ |
| 26A5 | $26$ | $117$ | $26$ | $25$ | $( 1,23, 9,21, 4,19,12,17, 7,15, 2,26,10,24, 5,22,13,20, 8,18, 3,16,11,14, 6,25)$ |
| 26A-5 | $26$ | $117$ | $26$ | $25$ | $( 1,22, 4,18, 7,14,10,23,13,19, 3,15, 6,24, 9,20,12,16, 2,25, 5,21, 8,17,11,26)$ |
| 39A1 | $13,3^{4},1$ | $78$ | $39$ | $20$ | $( 1,11, 8, 5, 2,12, 9, 6, 3,13,10, 7, 4)(14,18,17)(15,21,26)(16,24,22)(19,20,23)$ |
| 39A-1 | $13,3^{4},1$ | $78$ | $39$ | $20$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(14,18,17)(15,21,26)(16,24,22)(19,20,23)$ |
| 39A2 | $13,3^{4},1$ | $78$ | $39$ | $20$ | $( 1,11, 8, 5, 2,12, 9, 6, 3,13,10, 7, 4)(14,17,18)(15,26,21)(16,22,24)(19,23,20)$ |
| 39A-2 | $13,3^{4},1$ | $78$ | $39$ | $20$ | $( 1, 6,11, 3, 8,13, 5,10, 2, 7,12, 4, 9)(14,18,17)(15,21,26)(16,24,22)(19,20,23)$ |
| 39A4 | $13,3^{4},1$ | $78$ | $39$ | $20$ | $( 1,10, 6, 2,11, 7, 3,12, 8, 4,13, 9, 5)(14,17,18)(15,26,21)(16,22,24)(19,23,20)$ |
| 39A-4 | $13,3^{4},1$ | $78$ | $39$ | $20$ | $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)(14,17,18)(15,26,21)(16,22,24)(19,23,20)$ |
| 39A8 | $13,3^{4},1$ | $78$ | $39$ | $20$ | $( 1, 7,13, 6,12, 5,11, 4,10, 3, 9, 2, 8)(14,26,17)(15,22,20)(16,18,23)(21,24,25)$ |
| 39A-8 | $13,3^{4},1$ | $78$ | $39$ | $20$ | $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)(14,17,26)(15,20,22)(16,23,18)(21,25,24)$ |
Malle's constant $a(G)$: $1/8$
magma: ConjugacyClasses(G);
Character table
35 x 35 character tablemagma: CharacterTable(G);
Regular extensions
Data not computed